Problem: In writing the integers from 10 through 99 inclusive, how many times is the digit 6 written?
Explanation: The 9 integers 16, 26, $\ldots$, 86 and 96 each have 6 as a units digit.  The 10 integers 60, 61, $\ldots$, 68 and 69 each have 6 as a tens digit.  In total, there are $10+9=\boxed{19}$ appearances of the digit 6.

Note: This question asks for the number of times the digit 6 is written, not the number of numbers with the digit 6.